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On level crossings for a general class of piecewise-deterministic Markov processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

K. Borovkov  and
G. Last
K. Borovkov*
Affiliation:
University of Melbourne
G. Last*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address: k.borovkov@ms.unimelb.edu.au
∗∗ Postal address: Institut für Stochastik, Universität Karlsruhe (TH), D-76128 Karlsruhe, Germany. Email address: last@math.uni-karlsruhe.de
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Abstract

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We consider a piecewise-deterministic Markov process (Xt) governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. The paper deals with the point process of upcrossings of some level b by (Xt). We prove a version of Rice's formula relating the stationary density of (Xt) to level crossing intensities and show that, for a wide class of processes (Xt), as b → ∞, the scaled point process where ν+(b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process.

Keywords

Level crossingRice's formulacompound Poisson limit theorempiecewise-deterministic Markov process

MSC classification

Primary: 60J75: Jump processes 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 815 - 834
Copyright
Copyright © Applied Probability Trust 2008 

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